Dubinin isotherm

Is the X Plot Compatible with the Freundlich and Dubinin Isotherms ... 

The first term on the right is the common inverse cube law, the second is taken to be the empirically more important form for moderate film thickness (and also conforms to the polarization model, Section XVII-7C), and the last term allows for structural perturbation in the adsorbed film relative to bulk liquid adsorbate. In effect, the vapor pressure of a thin multilayer film is taken to be P and to relax toward P as the film thickens. The equation has been useful in relating adsorption isotherms to contact angle behavior (see Section X-7). Roy and Halsey [73] have used a similar equation earlier, Halsey [74] allowed for surface heterogeneity by assuming a distribution of Uq values in Eq. XVII-79. Dubinin s equation (Eq. XVII-75) has been mentioned another variant has been used by Bonnetain and co-workers [7S]. 

Adsorption isotherms in the micropore region may start off looking like one of the high BET c-value curves of Fig. XVII-10, but will then level off much like a Langmuir isotherm (Fig. XVII-3) as the pores fill and the surface area available for further adsorption greatly diminishes. The BET-type equation for adsorption limited to n layers (Eq. XVII-65) will sometimes fit this type of behavior. Currently, however, more use is made of the Dubinin-Raduschkevich or DR equation. Tliis is Eq. XVII-75, but now put in the form... 

These various considerations led Pierce, Wiley and Smith in 1949, and independently, Dubinin, to postulate that in very fine pores the mechanism of adsorption is pore filling rather than surface coverage. Thus the plateau of the Type 1 isotherm represents the filling up of the pores with adsorbate by a process similar to but not identical with capillary condensation, rather than a layer-by-layer building up of a film on the pore walls. 

An example from Dubinin s paper illustrates the application of the new equation. For benzene at 293 K, the plot of Equation (4.18) with m = 2 gave a line concave to the log (p°/p) axis. A revised value of m was therefore required, and was obtained as follows. A provisional value of IVq (= 408 mm g ) was estimated from the plateau of the isotherm the relative pressure (p/p°), corresponding to IV/tV = 0-368 was then read off, and a first value of S calculated from... 

Not all of the isotherm models discussed in the following are rigorous in the sense of being thermodynamically consistent. For example, specific deficiencies in the Freundhch, Sips, Dubinin-Radushkevich, Toth, and vacancy solution models have been identified (14). 

The n-butane isotherms shown in Fig. 8, in the concentration range up to about 2 per cent n-butane, can be expressed by the Dubinin-Astakov equation [34] ... 

The curve showing as a function of the relative pressure is called the isotherm, since the measurement is conducted at a constant temperature. Fig. 3.42 shows an adsorption isotherm of a so-called type IV (see Dubinin and Astakhov, 1971) isotherm material. This type of isotherm is very common. 

Dual nickel, 9 820—821 Dual-pressure processes, in nitric acid production, 17 175, 177, 179 Dual-solvent fractional extraction, 10 760 Dual Ziegler catalysts, for LLDPE production, 20 191 Dubinin-Radushkevich adsorption isotherm, 1 626, 627 Dubnium (Db), l 492t Ductile (nodular) iron, 14 522 Ductile brittle transition temperature (DBTT), 13 487 Ductile cast iron, 22 518—519 Ductile fracture, as failure mechanism, 26 983 Ductile iron... 

Porous texture characterization of all the samples was performed by physical adsorption of N2 at 77K. and CO2 at 273K, using an automatic adsorption system (Autosorb-6, Quantachrome). The micropore volume, Vpp (N2), was determined by application of Dubinin-Radushkevich equation to the N2 adsorption isotherm at 77K up to P/Po< 0.1. The volume of narrow micropores, Vnpp (DR,C02>, (mean pore size lower than 0.7 nm) was calculated from CO2 adsorption at 273 K. 

Mikhail, Brunauer, and Bodor proposed an extension of deBoer s r-method for the analysis of micropores which offers several advantages. These include the ability to obtain the micropore volume, surface, and their distributions from one experimental isotherm. Data for the MP (micropore analysis) method need not be measured at the very low pressures needed for the Dubinin and Kaganer theories. The method... 

Abstract To design an adsorption cartridge, it is necessary to be able to predict the service life as a function of several parameters. This prediction needs a model of the breakthrough curve of the toxic from the activated carbon bed. The most popular equation is the Wheeler-Jonas equation. We study the properties of this equation and show that it satisfies the constant pattern behaviour of travelling adsorption fronts. We compare this equation with other models of chemical engineering, mainly the linear driving force (LDF) approximation. It is shown that the different models lead to a different service life. And thus it is very important to choose the proper model. The LDF model has more physical significance and is recommended in combination with Dubinin-Radushkevitch (DR) isotherm even if no analytical solution exists. A numerical solution of the system equation must be used. 

This equation is different from the Wheeler equation. The first term on the right-hand side is identical and is the stoichiometric time t, but the second term includes the Langmuir coefficient K explicitly and in R. Thus no link with the Wheeler equation can be found. In addition this equation is valid solely with the Langmuir isotherm. This is a serious limitation because it has been recognized that Dubinin-Radushkevich (DR) approach is very useful. No analytical solution exists for the particular case of DR equation. A solution to this problem is to solve the system of equations by numerical methods. 

Dubinin-Raduskevish (DR) Due to the great significance of vapor adsorption on activated carbons, the DR isotherm, which is the most widely used in such cases, will be presented. It has several advantages (Dubinin, 1966) ... 

In this section, the most important parameters for the Dubinin-Raduskevish isotherm are presented. 

Comparison Between the Cohen-Kisarov and Dubinin-Radushkevich Equations. In a plot of log q vs. e2 the experimental points for one adsorption isotherm on zeolite frequently do not give a straight line, which would verify the Dubinin-Radushkevich equation. In this case, two distinct lines of different slopes are found (4). 

Let us assume that an experimental isotherm is perfectly described by the Cohen-Kisarov equation. When plotting the experimental points with the previous coordinates, three different cases may occur (1) if cmlA < 2, a case which was not yet found ((4) and Table I), the curve exhibits a constant convex curvature towards the ordinate axis (2) if cm /A > 2, the curve exhibits two distinct inflection points (Figure 3) where the experimental curve may easily be confused with the tangent to the inflection point, thus explaining the previous observations (3) if cm /A decreases to a value of 2, these inflection points are unified to give a large linear section, and the Dubinin-Radushkevich equation behaves as a limiting case of the Cohen-Kisarov equation. 

The fractional pore filling of the micropores of radius r at a given pressure P is given by the Dubinin-Radushkevich (DR) isotherm... 

Semiernpirical Isotherm Models. Some of these models have been shown to have some thermodynamic inconsistencies and should be used with due care. They include models based on the Polanyi adsorption potential (Dubinin-Radushkevich, Dubinin-Astakhov, Radke-Prausnitz, Toth, UNI LAN. and BET). 

Experimental isotherms from Barrer and Stuart (15) are used as a check on the present theory. Following the liquid filling theory of Dubinin (ll) and other (12, 13, lU), saturation values of for Argon on LiX at the various experimental temperatures are as follows ... 

On the other hand, for the microporous carbons with pore size distribution (PSD) with pore fractality, the pore fractal dimensions56,59,62 which represent the size distribution irregularity can be theoretically calculated by non-linear fitting of experimental adsorption isotherm with Dubinin-Astakhov (D-A) equation in consideration of PSD with pore fractality.143"149 The image analysis method54,151"153 has proven to be also effective for the estimation of the surface fractal dimension of the porous materials using perimeter-area method.154"159... 

Useful information about micropore structures can be derived from nitrogen or argon isotherm data in terms of the C-constant (BET), t or as-plots and the Dubinin-Radushkevich models. 

As described earlier, one of the first methods used to obtain PSD from the Dubinin equation is the so-called Dubinin-Stoeckli method [38-43], For strongly activated carbons with a heterogeneous collection of micropores, the overall adsorption isotherm is considered as a convolution of contributions from individual pore groups. Integrating the summation and assuming a normal Gaussian equation for the distribution of MPV with respect to the K parameter (Equation 4.19), Stoeckli obtained an equation useful to estimate the micro-PSD. 

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